Cat and Mouse
June 5, 2019 5:19 AM Subscribe
A cat is chasing a mouse which is trapped in a circular pond. The cat can run around the outside of the pond four times faster than the mouse can swim. Can the mouse escape?
Courtesy of Numberphile.
Courtesy of Numberphile.
First, assume a circular pond...
posted by St. Oops at 6:15 AM on June 5, 2019 [11 favorites]
posted by St. Oops at 6:15 AM on June 5, 2019 [11 favorites]
Is the cat on a conveyer belt?
obligatory
posted by thelonius at 6:23 AM on June 5, 2019 [11 favorites]
obligatory
posted by thelonius at 6:23 AM on June 5, 2019 [11 favorites]
It’s cute, although I suspect that if you give the cat some strategies other than "always stay as close to the mouse as possible" it gets more complex.
posted by Tell Me No Lies at 6:23 AM on June 5, 2019
posted by Tell Me No Lies at 6:23 AM on June 5, 2019
First, assume a circular pond...
If you don't do this then the pond's coastline is actually infinitely long so it will take the cat forever to get anywhere around it.
posted by Space Coyote at 6:24 AM on June 5, 2019 [24 favorites]
If you don't do this then the pond's coastline is actually infinitely long so it will take the cat forever to get anywhere around it.
posted by Space Coyote at 6:24 AM on June 5, 2019 [24 favorites]
Assume cat is not a cheetah.
posted by romanb at 6:40 AM on June 5, 2019 [5 favorites]
posted by romanb at 6:40 AM on June 5, 2019 [5 favorites]
It’s cute, although I suspect that if you give the cat some strategies other than "always stay as close to the mouse as possible" it gets more complex.
Once the mouse gets to the sweet spot circle (cat can do whatever it likes while this is happening) the mouse can always make the angle sufficiently close to 180 degrees to escape.
posted by Obscure Reference at 6:47 AM on June 5, 2019 [4 favorites]
Once the mouse gets to the sweet spot circle (cat can do whatever it likes while this is happening) the mouse can always make the angle sufficiently close to 180 degrees to escape.
posted by Obscure Reference at 6:47 AM on June 5, 2019 [4 favorites]
It’s cute, although I suspect that if you give the cat some strategies other than "always stay as close to the mouse as possible" it gets more complex.
The point of the mouse's strategy is that it always allows the mouse to get 180° around the pond from the cat while being close enough to shore to escape, no matter what the cat does. If the cat did something other than "stay as close as possible to the mouse" while the mouse was swimming around, then the angle between the mouse and the cat would just get up to 180° faster, and then the mouse can escape.
posted by Johnny Assay at 6:48 AM on June 5, 2019 [7 favorites]
The point of the mouse's strategy is that it always allows the mouse to get 180° around the pond from the cat while being close enough to shore to escape, no matter what the cat does. If the cat did something other than "stay as close as possible to the mouse" while the mouse was swimming around, then the angle between the mouse and the cat would just get up to 180° faster, and then the mouse can escape.
posted by Johnny Assay at 6:48 AM on June 5, 2019 [7 favorites]
Numberphile videos are usually safe to read the comments wherein I learned that the ratio between cat an mouse where the cat can always get the mouse is pi+1 (i.e. close to 4),
posted by Obscure Reference at 6:52 AM on June 5, 2019 [3 favorites]
posted by Obscure Reference at 6:52 AM on June 5, 2019 [3 favorites]
A version of this puzzle also showed up in the FiveThirtyEight Riddler column a few years back. (Answers were in the following week's column.) The mouse's strategy in the video works for a cat that can run up to π + 1 ≈ 4.1416 times faster than the mouse can swim; but it turns out that with a little additional cleverness, the mouse can escape a cat that can run ≈ 4.6033 times faster than it can swim.
posted by Johnny Assay at 6:56 AM on June 5, 2019 [5 favorites]
posted by Johnny Assay at 6:56 AM on June 5, 2019 [5 favorites]
Cats have reach, and they are not shy about getting a paw wet.
posted by Meatbomb at 7:00 AM on June 5, 2019 [3 favorites]
posted by Meatbomb at 7:00 AM on June 5, 2019 [3 favorites]
i.e., the actual "safe zone" of the pond is the radius minus the cat's full extension, with it's center of gravity just as far as possible into the pond to maintain balance on the shoreline.
posted by Meatbomb at 7:01 AM on June 5, 2019 [2 favorites]
posted by Meatbomb at 7:01 AM on June 5, 2019 [2 favorites]
This is some weird Voight-Kampff bullshit.
"If the mouse has to keep swimming, it will eventually get tired and drown. Why aren't you helping the mouse, Leon?"
posted by TheWhiteSkull at 7:04 AM on June 5, 2019 [43 favorites]
"If the mouse has to keep swimming, it will eventually get tired and drown. Why aren't you helping the mouse, Leon?"
posted by TheWhiteSkull at 7:04 AM on June 5, 2019 [43 favorites]
According to speedofanimals.com, cats can run at 48km/h (which, in some kind of animal-speed wind chill calculation apparently "feels like" 173km/h). So this mouse is swimming at 12 km/h, or in nautical terms 6.5 knots. If it can swim so well, it probably also has the option of just staying in the pool and adapting to its new life as a sea-mouse.
posted by sfenders at 7:07 AM on June 5, 2019 [41 favorites]
posted by sfenders at 7:07 AM on June 5, 2019 [41 favorites]
Well, if we're taking this apart like that, the cat's top speed is less relevant than its ability to accelerate, decelerate, and turn, and its grip against lateral g-loading. In the model as presented, the cat is basically [terrible 80s reference] Automan [/80s], making instantaneous changes in speed and direction.
posted by GCU Sweet and Full of Grace at 7:17 AM on June 5, 2019 [9 favorites]
posted by GCU Sweet and Full of Grace at 7:17 AM on June 5, 2019 [9 favorites]
>> It’s cute, although I suspect that if you give the cat some
>> strategies other than "always stay as close to the mouse as possible"
>> it gets more complex.
>
> Once the mouse gets to the sweet spot circle (cat can do whatever it likes
> while this is happening) the mouse can always make the angle sufficiently
> close to 180 degrees to escape.
That seems intuitively clear. However, I am also aware that the intuitively satisfying "mouse stays as far away as possible from the cat" strategy fails. Hence my suspicion.
posted by Tell Me No Lies at 7:45 AM on June 5, 2019
>> strategies other than "always stay as close to the mouse as possible"
>> it gets more complex.
>
> Once the mouse gets to the sweet spot circle (cat can do whatever it likes
> while this is happening) the mouse can always make the angle sufficiently
> close to 180 degrees to escape.
That seems intuitively clear. However, I am also aware that the intuitively satisfying "mouse stays as far away as possible from the cat" strategy fails. Hence my suspicion.
posted by Tell Me No Lies at 7:45 AM on June 5, 2019
sfenders: "So this mouse is swimming at 12 km/h, or in nautical terms 6.5 knots."
HOW MANY FAHRENHEIT ARE IN A CUP
posted by signal at 7:55 AM on June 5, 2019 [10 favorites]
HOW MANY FAHRENHEIT ARE IN A CUP
posted by signal at 7:55 AM on June 5, 2019 [10 favorites]
sfenders: which, in some kind of animal-speed wind chill calculation apparently "feels like" 173km/h
Is that for a short-hair cat or a long-hair?
posted by clawsoon at 7:58 AM on June 5, 2019 [1 favorite]
Is that for a short-hair cat or a long-hair?
posted by clawsoon at 7:58 AM on June 5, 2019 [1 favorite]
It could grip it by the husk.
posted by RobotVoodooPower at 8:00 AM on June 5, 2019 [15 favorites]
posted by RobotVoodooPower at 8:00 AM on June 5, 2019 [15 favorites]
Not enough information to answer the question in the OP.
Assuming a constant-speed mouse once out of the pond, we need that speed and the set of points near the pond where the mouse can be considered to have evaded the cat once reached.
posted by "mad dan" eccles at 8:11 AM on June 5, 2019 [1 favorite]
Assuming a constant-speed mouse once out of the pond, we need that speed and the set of points near the pond where the mouse can be considered to have evaded the cat once reached.
posted by "mad dan" eccles at 8:11 AM on June 5, 2019 [1 favorite]
HOW MANY FAHRENHEIT ARE IN A CUP
I'm measuring my speed in attoparsec per microfortnight, thank you very much.
posted by DreamerFi at 8:15 AM on June 5, 2019 [4 favorites]
I'm measuring my speed in attoparsec per microfortnight, thank you very much.
posted by DreamerFi at 8:15 AM on June 5, 2019 [4 favorites]
Assume cat is not a cheetah.
The video tells us up front: this cat is playing by the rules.
I'll see myself out
posted by Mayor West at 8:36 AM on June 5, 2019 [20 favorites]
The video tells us up front: this cat is playing by the rules.
I'll see myself out
posted by Mayor West at 8:36 AM on June 5, 2019 [20 favorites]
My thought was "mouse escapes cat by being eaten by a bass", but I like sfenders sea-mouse option better.
posted by coppertop at 8:38 AM on June 5, 2019 [3 favorites]
posted by coppertop at 8:38 AM on June 5, 2019 [3 favorites]
First, assume a circular pond...
And a frictionless cat.
posted by carter at 9:00 AM on June 5, 2019 [9 favorites]
And a frictionless cat.
posted by carter at 9:00 AM on June 5, 2019 [9 favorites]
"How long will this mouse keep treading water before giving up and drowning" is an animal model of depression — antidepressants make them hold out longer.
Whch makes sfenders' Sea Mouse, like, the animal patron saint of full recovery and triumph over mental illness, right?
posted by nebulawindphone at 9:08 AM on June 5, 2019 [2 favorites]
Whch makes sfenders' Sea Mouse, like, the animal patron saint of full recovery and triumph over mental illness, right?
posted by nebulawindphone at 9:08 AM on June 5, 2019 [2 favorites]
I suspect the optimal strategy (in terms of escape time) is "swim towards edge opposite from cat until reaching safe radius, then dash" ... that would make use of the better angular velocity ($\omega_{Mouse}$) close to the center
posted by kleinsteradikaleminderheit at 9:41 AM on June 5, 2019 [1 favorite]
posted by kleinsteradikaleminderheit at 9:41 AM on June 5, 2019 [1 favorite]
Which of these would be most distracted by the plate of beans?
posted by OHenryPacey at 10:14 AM on June 5, 2019 [1 favorite]
posted by OHenryPacey at 10:14 AM on June 5, 2019 [1 favorite]
I have no idea how these people got their mice wedged into their ponds, or why.
posted by blurker at 10:27 AM on June 5, 2019 [3 favorites]
posted by blurker at 10:27 AM on June 5, 2019 [3 favorites]
I'm glad to see Numberphile here on the blue. It's always great seeing people share what they love and making it accessible to non-specialists.
posted by sjswitzer at 10:30 AM on June 5, 2019 [1 favorite]
posted by sjswitzer at 10:30 AM on June 5, 2019 [1 favorite]
I'm not sure about the J-shaped strategy that leads to the 4.6 value from Johnny Assay's link. The strategy hinges on the argument that it doesn't make sense for the cat/dog/monster to change directions once the mouse/duck/rower begins the dash, but I don't think that's true. At the exact moment the mouse and cat are 180 degrees out of phase, the system is symmetric. No constraint is placed on the cat's acceleration or reaction time, so it can immediately change directions and easily catch the mouse. The mouse could instead take an h-shaped path by going the opposite direction on the tangent when starting its dash but then the cat just continues its path and catches the mouse. Without working out the math further, I think symmetry suggests that the radial dash really is optimal and 4.14 is the fastest the cat can be for the mouse to escape.
posted by biogeo at 10:31 AM on June 5, 2019
posted by biogeo at 10:31 AM on June 5, 2019
I was initially interested in this, but coming straight from the YouTube hate speech thread, it felt pretty gross to watch a YT video.
posted by nickmark at 10:37 AM on June 5, 2019 [3 favorites]
posted by nickmark at 10:37 AM on June 5, 2019 [3 favorites]
As the former owner of five Maine Coons, I would say that any one of those monsters would have just waded in and retrieved the mouse.
posted by Ber at 10:42 AM on June 5, 2019 [3 favorites]
posted by Ber at 10:42 AM on June 5, 2019 [3 favorites]
Totally different math if it's a mouse and a flerken.
posted by otherchaz at 10:50 AM on June 5, 2019
posted by otherchaz at 10:50 AM on June 5, 2019
Why aren't you helping the mouse, Leon?
“Because I am a Nexus 6. And a cat.”
posted by GenjiandProust at 11:07 AM on June 5, 2019 [1 favorite]
“Because I am a Nexus 6. And a cat.”
posted by GenjiandProust at 11:07 AM on June 5, 2019 [1 favorite]
So this mouse is swimming at 12 km/h, or in nautical terms 6.5 knots.
Just to clarify for anybody who's having trouble visualizing how fast that is, it's awfully freakin' fast for a mouse to be swimming. It's also faster than you're allowed to go in a no-wake zone (5 knots) so depending on where he is, the mouse may get in trouble with the harbormaster.
posted by mstokes650 at 11:28 AM on June 5, 2019 [9 favorites]
Just to clarify for anybody who's having trouble visualizing how fast that is, it's awfully freakin' fast for a mouse to be swimming. It's also faster than you're allowed to go in a no-wake zone (5 knots) so depending on where he is, the mouse may get in trouble with the harbormaster.
posted by mstokes650 at 11:28 AM on June 5, 2019 [9 favorites]
What if it's a Turkish Van cat (which apparently loves swimming)?
posted by caution live frogs at 11:45 AM on June 5, 2019 [1 favorite]
posted by caution live frogs at 11:45 AM on June 5, 2019 [1 favorite]
Does the mouse have a frickin' laser beam attached to its head?
posted by It's Raining Florence Henderson at 12:04 PM on June 5, 2019 [1 favorite]
posted by It's Raining Florence Henderson at 12:04 PM on June 5, 2019 [1 favorite]
If the mouse can evolve into a godlike techno-hyperrodent it can develop a means of altering the speed of light in the vicinity of the pond, thus constraining the cat's maximum velocity and becoming capable of escaping from any cat no matter how fast it can run. Also, from cats which can transform into black holes.
posted by XMLicious at 12:07 PM on June 5, 2019 [2 favorites]
posted by XMLicious at 12:07 PM on June 5, 2019 [2 favorites]
Only vaguely relevant, but on the subject of numberphile, this card trick is maths witchcraft
posted by juv3nal at 12:32 PM on June 5, 2019 [3 favorites]
posted by juv3nal at 12:32 PM on June 5, 2019 [3 favorites]
Problem #2: If the beeline-feline continues to move in a straight line towards the mouse after it exits the pond, how far must the mouse run before it can safely get an hour of sleep?
posted by oulipian at 1:05 PM on June 5, 2019 [2 favorites]
posted by oulipian at 1:05 PM on June 5, 2019 [2 favorites]
Can someone summarize this for those who don't have time to watch a 20 minute video?
posted by iamnotangry at 3:40 PM on June 5, 2019 [1 favorite]
posted by iamnotangry at 3:40 PM on June 5, 2019 [1 favorite]
Can someone summarize this for those who don't have time to watch a 20 minute video?
The obvious solution is for the mouse to stand at the center of the circle and then run like hell directly away from the cat. This won't work because the cat can circle the pond faster than the mouse can run to the edge.
The mouse needs to start its run from closer to the edge of the pond to succeed. It can do this by running in a small circle around the center. Running in a circle lures the cat progressively further out of position; when the cat is fully out of position the mouse can make the dash from the edge of the circle it's been running to the edge of the pond.
posted by Tell Me No Lies at 3:54 PM on June 5, 2019 [7 favorites]
The obvious solution is for the mouse to stand at the center of the circle and then run like hell directly away from the cat. This won't work because the cat can circle the pond faster than the mouse can run to the edge.
The mouse needs to start its run from closer to the edge of the pond to succeed. It can do this by running in a small circle around the center. Running in a circle lures the cat progressively further out of position; when the cat is fully out of position the mouse can make the dash from the edge of the circle it's been running to the edge of the pond.
posted by Tell Me No Lies at 3:54 PM on June 5, 2019 [7 favorites]
The only things I would add to Tell Me No Lies' excellent summary are: first, pi is involved; and second, it's very important that the cat is exactly four times faster than the mouse because it just barely works for the mouse at that ratio but it doesn't for anything more.
posted by yhbc at 10:46 PM on June 5, 2019 [1 favorite]
posted by yhbc at 10:46 PM on June 5, 2019 [1 favorite]
This video is very mouse-centric. I would like to help the cat. The solution of course is, assume 2 cats. /not mouseist
posted by chavenet at 2:13 AM on June 6, 2019
posted by chavenet at 2:13 AM on June 6, 2019
Without working out the math further, I think symmetry suggests that the radial dash really is optimal
That symmetry is unstable, though, and it breaks once the mouse is more than the critical radius from the centre.
Let R be the ratio of the cat's running speed to the mouse's swimming speed, and let the radius of the pond be normalized to 1. The mouse's initial strategy, then, is to swim to a point M, on the edge of a circle of radius 1/R that's 180° away from the cat at point C on the edge, which it can always do regardless of R.
At this point the mouse starts swimming straight toward a point P at the edge of the pond, 180° away from the cat; instantaneously, the cat picks a direction and starts running around the edge toward P and again instantaneously, the mouse adjusts its own direction and starts swimming straight toward point Q, further around the pond in the opposite direction from the cat, instead.
If the cat now reverses direction, the mouse can also revise its own: first swinging back toward P again and then, once the cat has passed the 180° mark behind it, toward Q', a little further around the edge in the opposite direction from the cat's new path. These direction adjustments will be quite small, and will add much less length (proportionally) to the mouse's path than the cat's; and there will come a time when the cat and the mouse will once again be 180° apart but the mouse will be further out from the centre than 1/R, making it clear that the cat has wasted its time. So that won't happen, and we can forget about direction reversals and Q', and concentrate on working out where Q has to be.
posted by flabdablet at 5:01 AM on June 6, 2019 [2 favorites]
That symmetry is unstable, though, and it breaks once the mouse is more than the critical radius from the centre.
Let R be the ratio of the cat's running speed to the mouse's swimming speed, and let the radius of the pond be normalized to 1. The mouse's initial strategy, then, is to swim to a point M, on the edge of a circle of radius 1/R that's 180° away from the cat at point C on the edge, which it can always do regardless of R.
At this point the mouse starts swimming straight toward a point P at the edge of the pond, 180° away from the cat; instantaneously, the cat picks a direction and starts running around the edge toward P and again instantaneously, the mouse adjusts its own direction and starts swimming straight toward point Q, further around the pond in the opposite direction from the cat, instead.
If the cat now reverses direction, the mouse can also revise its own: first swinging back toward P again and then, once the cat has passed the 180° mark behind it, toward Q', a little further around the edge in the opposite direction from the cat's new path. These direction adjustments will be quite small, and will add much less length (proportionally) to the mouse's path than the cat's; and there will come a time when the cat and the mouse will once again be 180° apart but the mouse will be further out from the centre than 1/R, making it clear that the cat has wasted its time. So that won't happen, and we can forget about direction reversals and Q', and concentrate on working out where Q has to be.
posted by flabdablet at 5:01 AM on June 6, 2019 [2 favorites]
This is some weird Voight-Kampff bullshit.
"If the mouse has to keep swimming, it will eventually get tired and drown. Why aren't you helping the mouse, Leon?"
“They're just questions, Leon. In answer to your query, they're written down for me. It's a test designed to provoke an emotional response.
Now describe, in simple words, only the good things that come to your mind when you think about your litter mates.”
/sound of angry yowling, repeated vicious scratching
posted by Ghidorah at 8:07 AM on June 6, 2019
"If the mouse has to keep swimming, it will eventually get tired and drown. Why aren't you helping the mouse, Leon?"
“They're just questions, Leon. In answer to your query, they're written down for me. It's a test designed to provoke an emotional response.
Now describe, in simple words, only the good things that come to your mind when you think about your litter mates.”
/sound of angry yowling, repeated vicious scratching
posted by Ghidorah at 8:07 AM on June 6, 2019
I'm not sure about the J-shaped strategy that leads to the 4.6 value from Johnny Assay's link. The strategy hinges on the argument that it doesn't make sense for the cat/dog/monster to change directions once the mouse/duck/rower begins the dash, but I don't think that's true. At the exact moment the mouse and cat are 180 degrees out of phase, the system is symmetric.
Another way to think about this: there are a whole pile of stable states where the mouse and cat are 180° out of phase that are not clockwise/anticlockwise symmetric because both mouse and cat have a velocity in one particular direction, either clockwise or anticlockwise.
That asymmetry doesn't go away even if you allow instantaneous direction reversals, because the mirror image of each of these asymmetric states is also asymmetric.
The only fully symmetric stable state is where both those velocities are zero, i.e. the cat is at rest on the edge of the pond and the mouse is at rest at a point on a critical circle 180° away from it.
If the cat allows the system to reach such a state, the mouse's best strategy is to begin a very slow drift directly away from the cat. It will take the cat a bit of time to figure out that the mouse is actually moving, at which point the mouse is now on the edge of a circle bigger than the one it started on. As soon as the cat does move, the mouse makes an immediate 90° turn away from it and starts swimming at top speed, in a straight line tangential to this new, bigger circle.
The cat won't reverse direction again because it's been steadily gaining on the mouse since taking off, but the mouse is better off than it would have been had it started its tangential run from the original critical circle. It might even beat a 4.6x cat this way.
From which it follows that it's not in the cat's best interest to allow such a zero-speed symmetric state, or anything even close to it, to occur; its optimal strategy is to keep running around the edge of the pond at top speed, in order to force the mouse to begin its tangential dash from 180° away on as small a circle as possible.
posted by flabdablet at 9:06 AM on June 6, 2019
Another way to think about this: there are a whole pile of stable states where the mouse and cat are 180° out of phase that are not clockwise/anticlockwise symmetric because both mouse and cat have a velocity in one particular direction, either clockwise or anticlockwise.
That asymmetry doesn't go away even if you allow instantaneous direction reversals, because the mirror image of each of these asymmetric states is also asymmetric.
The only fully symmetric stable state is where both those velocities are zero, i.e. the cat is at rest on the edge of the pond and the mouse is at rest at a point on a critical circle 180° away from it.
If the cat allows the system to reach such a state, the mouse's best strategy is to begin a very slow drift directly away from the cat. It will take the cat a bit of time to figure out that the mouse is actually moving, at which point the mouse is now on the edge of a circle bigger than the one it started on. As soon as the cat does move, the mouse makes an immediate 90° turn away from it and starts swimming at top speed, in a straight line tangential to this new, bigger circle.
The cat won't reverse direction again because it's been steadily gaining on the mouse since taking off, but the mouse is better off than it would have been had it started its tangential run from the original critical circle. It might even beat a 4.6x cat this way.
From which it follows that it's not in the cat's best interest to allow such a zero-speed symmetric state, or anything even close to it, to occur; its optimal strategy is to keep running around the edge of the pond at top speed, in order to force the mouse to begin its tangential dash from 180° away on as small a circle as possible.
posted by flabdablet at 9:06 AM on June 6, 2019
I would love to play with the GeoGebra simulation in this video without having to re-write it myself.
posted by fantabulous timewaster at 2:28 PM on June 7, 2019
posted by fantabulous timewaster at 2:28 PM on June 7, 2019
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posted by starfishprime at 6:06 AM on June 5, 2019